\[ \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx \]
Optimal antiderivative \[ -\frac {2 \left (36 A a b -24 B \,a^{2}-49 b^{2} B \right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{315 b^{3} d}+\frac {2 \left (3 A b -2 B a \right ) \cos \left (d x +c \right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{21 b^{2} d}+\frac {2 B \left (\cos ^{2}\left (d x +c \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{9 b d}+\frac {2 \left (24 A \,a^{2} b +75 A \,b^{3}-16 a^{3} B -36 B a \,b^{2}\right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{315 b^{3} d}+\frac {2 \left (24 A \,a^{3} b +57 A a \,b^{3}-16 a^{4} B -24 B \,a^{2} b^{2}+147 b^{4} B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{315 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}-\frac {2 \left (a^{2}-b^{2}\right ) \left (24 A \,a^{2} b +75 A \,b^{3}-16 a^{3} B -36 B a \,b^{2}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{315 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (-32 i \, B a^{5} + 48 i \, A a^{4} b - 36 i \, B a^{3} b^{2} + 96 i \, A a^{2} b^{3} - 39 i \, B a b^{4} - 225 i \, A b^{5}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + \sqrt {2} {\left (32 i \, B a^{5} - 48 i \, A a^{4} b + 36 i \, B a^{3} b^{2} - 96 i \, A a^{2} b^{3} + 39 i \, B a b^{4} + 225 i \, A b^{5}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) - 3 \, \sqrt {2} {\left (16 i \, B a^{4} b - 24 i \, A a^{3} b^{2} + 24 i \, B a^{2} b^{3} - 57 i \, A a b^{4} - 147 i \, B b^{5}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {2} {\left (-16 i \, B a^{4} b + 24 i \, A a^{3} b^{2} - 24 i \, B a^{2} b^{3} + 57 i \, A a b^{4} + 147 i \, B b^{5}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) + 6 \, {\left (35 \, B b^{5} \cos \left (d x + c\right )^{3} + 8 \, B a^{3} b^{2} - 12 \, A a^{2} b^{3} + 13 \, B a b^{4} + 75 \, A b^{5} + 5 \, {\left (B a b^{4} + 9 \, A b^{5}\right )} \cos \left (d x + c\right )^{2} - {\left (6 \, B a^{2} b^{3} - 9 \, A a b^{4} - 49 \, B b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{945 \, b^{5} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B \cos \left (d x + c\right )^{4} + A \cos \left (d x + c\right )^{3}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]